×

zbMATH — the first resource for mathematics

A symmetric solution of a multipoint boundary value problem at resonance. (English) Zbl 1137.34313
Summary: We apply a coincidence degree theorem of Mawhin to show the existence of at least one symmetric solution of the nonlinear second-order multipoint boundary value problem
\[ \begin{alignedat}{2} 2 u''(t)&= f(t,u(t),|u'(t)|), &\quad&t \in (0,1),\\ u(0)&= \sum_{i=1}^n \mu_i u(\xi_i),\;u(1-t) = u(t), &\quad &t \in [0,1], \end{alignedat} \]
where \(0 < \xi_1 <\xi_2 < \cdots < \xi_n \leq {1}/{2}\), \(\sum_{i=1}^n \mu_i = 1\), \(f: [0,1]\times \mathbb{R}^{2} \rightarrow \mathbb{R}\) with \(f(t,x,y) = f(1-t,x,y)\), \((t,x,y) \in [0,1]\times \mathbb{R}^{2}\), satisfying the Carathéodory conditions.

MSC:
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] W. Feng and J. R. L. Webb, “Solvability of three point boundary value problems at resonance,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, vol. 30, no. 6, pp. 3227-3238, 1997. · Zbl 0891.34019
[2] C. P. Gupta, “A second order m-point boundary value problem at resonance,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, vol. 24, no. 10, pp. 1483-1489, 1995. · Zbl 0824.34023
[3] C. P. Gupta, “Existence theorems for a second order m-point boundary value problem at resonance,” International Journal of Mathematics and Mathematical Sciences, vol. 18, no. 4, pp. 705-710, 1995. · Zbl 0839.34027
[4] C. P. Gupta, “Solvability of a multi-point boundary value problem at resonance,” Results in Mathematics. Resultate der Mathematik, vol. 28, no. 3-4, pp. 270-276, 1995. · Zbl 0843.34023
[5] B. Liu, “Solvability of multi-point boundary value problem at resonance. II,” Applied Mathematics and Computation, vol. 136, no. 2-3, pp. 353-377, 2003. · Zbl 1053.34016
[6] B. Liu, “Solvability of multi-point boundary value problem at resonance. IV,” Applied Mathematics and Computation, vol. 143, no. 2-3, pp. 275-299, 2003. · Zbl 1071.34014
[7] Y. Liu and W. Ge, “Solvability of a (P,N - P)-type multi-point boundary-value problem for higher-order differential equations,” Electronic Journal of Differential Equations, vol. 2003, no. 120, pp. 1-19, 2003. · Zbl 1042.34032
[8] B. Liu and J. S. Yu, “Solvability of multi-point boundary value problem at resonance. III,” Applied Mathematics and Computation, vol. 129, no. 1, pp. 119-143, 2002. · Zbl 1054.34033
[9] B. Liu and J. S. Yu, “Solvability of multi-point boundary value problems at resonance. I,” Indian Journal of Pure and Applied Mathematics, vol. 33, no. 4, pp. 475-494, 2002. · Zbl 1021.34013
[10] J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, vol. 40 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Rhode Island, 1979. · Zbl 0414.34025
[11] X. Ni and W. Ge, “Multi-point boundary-value problems for the p-Laplacian at resonance,” Electronic Journal of Differential Equations, vol. 2003, no. 112, pp. 1-7, 2003. · Zbl 1046.34031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.